Romance of Dimensions

This masterpiece of science (and mathematical) fiction is a delightfully unique and highly entertaining satire that has charmed readers for more than 100 years.

The work of English clergyman, educator and Shakespearean scholar Edwin A. Abbott (1838-1926), it describes the journeys of A. Square, a mathematician and resident of the two-dimensional Flatland, where women-thin, straight lines-are the lowliest of shapes, and where men may have any number of sides, depending on their social status.

The first half of the story goes through the practicalities of existing in a two-dimensional universe as well as a history leading up to the year 1999 on the eve of the 3rd Millennium.
On New Year's Eve, the Square dreams about a visit to a one-dimensional world (Lineland) inhabited by "lustrous points". These points are unable to see the Square as anything other than a set of points on a line. Thus, the Square attempts to convince the realm's monarch of a second dimension; but is unable to do so. In the end, the monarch of Lineland tries to kill A Square rather than tolerate his nonsense.

Following this vision, he is himself visited by a three-dimensional sphere. Similar to the "points" in Lineland, the Square is unable to see the sphere as anything other than a circle. The Sphere then levitates up and down through the Flatland, allowing Square to see the circle expand and retract. The Square is not fully convinced until he sees Spaceland (a tridimensional world) for himself. This Sphere visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a third dimension in the hope of eventually educating the population of Flatland. From the safety of Spaceland, they are able to observe the leaders of Flatland secretly acknowledging the existence of the sphere and prescribing the silencing of anyone found preaching the truth of Spaceland and the third dimension. After this proclamation is made, many witnesses are massacred or imprisoned, including A Square's brother, B.

Square tries to convince the Sphere of the theoretical possibility of the existence of a fourth and higher spatial dimensions; but the Sphere returns his student to Flatland in disgrace.

The Square then has a dream, which the Sphere visits him again, this time to introduce him to Pointland, whereof the point perceives any communication as a thought originating in his own mind: "You see," said my Teacher, "how little your words have done. So far as the Monarch understands them at all, he accepts them as his own – for he cannot conceive of any other except himself – and plumes himself upon the variety of Its Thought as an instance of creative Power. Let us leave this god of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction.

The Square recognizes the identity of the ignorance of the monarchs of Pointland and Lineland with his own (and the Sphere's) previous ignorance of the existence of higher dimensions. Once returned to Flatland, the Square cannot convince anyone of Spaceland's existence, especially after official decrees are announced that anyone preaching the existence of three dimensions will be imprisoned (or executed, depending on caste). Eventually the Square himself is imprisoned for just this reason, with only occasional contact with his brother who is imprisoned in the same facility. He does not manage to convince his brother, even after all they have both seen. Seven years after being imprisoned, A Square writes out the book Flatland in the form of a memoir, hoping to keep it as posterity for a future generation that can see beyond their two-dimensional existence.

Flatland is not only fascinating reading, it is still a first-rate fictional introduction to the concept of the multiple dimensions of space. "Instructive, entertaining, and stimulating to the imagination." — Mathematics Teacher.